[_8j50]

Ok, please ELI5 this for me, The very statement that anything at all is random is completley absurd to me, especially from an academic context. Are they using a definition of random that is equivalent to "nearly impossible to predict"? , I mean, yes, from the perspective of a limited observer random things can exist, but in an absolute sense, for something to be random then even with the knowledge of all things past that lead up to an event, you would not be able to find the cause of that event because it was truly random.

Do they mean "unpredictable for humans with known means of predicting"?

You might as well use the term "magical" or "miraculous" if you use "random".

[oh_my_goodness]

Lemme go through details, not because anyone doesn't know the details, but to get them written down and discussable.

"Random" means I draw up a list of all possible outcomes. The probability of an event is defined as the fraction of outcomes in which that event is true.

Example. Choose two numbers "randomly" between 1 and 3. What's the probability that the two numbers are equal? The list of outcomes is: (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3). There are 9 possible outcomes. In 3 outcomes the first and second numbers are equal. So the probability is 3/9 = 1/3.

The math part is just bookkeeping.* The math isn't random, it just uses the rule above. And there is no claim (within the math) that things are or aren't "really random" in the actual world.

*Can be very slippery bookkeeping. People who know what they're doing get wrong answers all the time. But bookkeeping.

[_8j50]

Thank you, my original comment was for ELI5, I was asking because I didn't know and clarifying like this helps.

Wouldn't it be more correct to say arbitrary instead of random?

[oh_my_goodness]

I think random is often used as a keyword that means "use ideas from probability." Those ideas are tricky and (I think) not well captured by any one word.

Example of a tricky situation: Say we play cards. You shuffle the cards well, and you don't show them to me. I might say the order of the deck is random, to indicate that I have no idea which card is in what position. But you might be looking at the faces of the cards. In that case you might say the same cards are ordered in a way that's not random.

Could we describe the same situation using the word arbitrary but not the word random? I'm not sure. Seems likely.

I know how to make up models and test the models by experiment. I don't know whether randomness is real or not. I don't think it's deeply meaningful what words we use, as long as it's clear what the math is and how we're applying it.

[_8j50]

I agree, but in crypto for example they use the term "cryptographically random" as in for the purposes of computing it is not possible to predict the value.

As a complete outsider to the field, I had no idea what is meant by random in a mathematical sense because I thought math was always realistic, as in it mirrors real things, you can't have 2+2=3 because reality doesn't work that way.

[d--b]

This is mathematics, not physics. You can define stuff that don't "really" exist in the world, but that work for the purpose of calculating stuff.

"randomness" has as much right to exist in a mathematical sense as a "point" or a "plane".

[thfuran]

Randomness exists in physics as well.

[d--b]

I am just responding to the comment which assumes that it doesn't.

[_8j50]

How can you prove something that isn't real? How is randomness defined as a concept in mathematics?

[siver_john]

As long as you can clearly define your starting axioms you can mathematically prove or disprove anything. The physical example would be to just say, suppose a universe with the same properties of ours exist but where the gravitational constant is twice as large prove whether X can occur in that universe.

As far as how randomness is defined I believe that might be field dependent but this is me getting out of my depth (I've only taken a few courses on combinatorics).

[DavidSJ]

Randomness is usually defined in terms of a probability distribution, which is just a measure where the total mass assigned to all elements adds up to 1.

[majkinetor]

You can actually only prove unreal things because you impose limitations. Real things can not be proven, but only disproved.

[rrajasek]

What they're using is the mathematical definition of randomness - an event whose outcomes cannot be determined prior to its occurence. The assumption is that it's physically impossible to predict with certainty what the true outcome is.

In many cases, what you're describing applies - there are many real world phenomena for which we have very limited information or are too complicated to model exactly. Probability theory is really useful in these situations. Why probability theory is useful is because even unpredictable events have some high level patterns that we can discover.

In the case of the article, they're interested in understanding properties of random graphs. Random graphs are useful because they're useful models of real life graphs (such as for social media websites). This is because the real life graph is constructed in a way that appears "random" (such as to whoever maintains the social media site). A lot of graph properties are proxies for real life social behavior - triangles indicate close knit groups, cycles indicate broad friend circles. If you can estimate some of these properties in a random graph, you can do the same for real life graphs. For e.g. Facebook might be interested in identifying close-knit friend circles to identify potential users who may be friends or recommend groups that users can join based on who their friends are. It will also give them sensible estimates on how many people know each other personally, how many friend circles exist and so on.

However, there are certain real life processes which are considered to be 'truly random' - quantum tunnelling, radioactive decay. It's physically impossible (regardless of what technology we can develop) to predict if an electron will tunnel through a barrier, or how long it takes for an uranium atom to decay. However, there are patterns to this randomness such as the probability that the particle tunnels and the half life for radioactive decay. These are macroscopic descriptions of random phenomena.

EDIT: added more relevant information about random graphs